Formal theories of fuzzy logic

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1. Introduction to the problem

Mathematical fuzzy logic is a special formal theory of many-valued logic generalizing classical mathematical logic that focuses on the development of tools for modeling vagueness phenomenon. We distinguish fuzzy logic in narrow (FLn) and in broader sense (FLb). Fuzzy logic has been initiated at the end of sixties and beginning of seventies by L. A. Zadeh and J. A. Goguen. Nowadays, it is a well developed theory thanks to the results of an international group of mathematicians (e.g., P. Hájek, D. Mundici, S. Gottwald, F. Esteva, L. Godo, A. diNola, F. Montagna, V. Novák, P. Cintula, L. Běhounek, and many others).

FLn provide fundamental formalism while FLb is its extension aiming at modeling human way of reasoning, important feature of which is the use of natural language. Hence, the necessary constituent of FLb is formalization of approximate reasoning – the reasoning realized by people on the basis of imprecise information formulated in natural language. Fuzzy logic puts forth special questions either not raised it classical logic, or having no sense in it. Typical problems of fuzzy logic is formalization of approximate equality, formalization of semantics of evaluative linguistic expressions, finding a conclusion on the basis of the assumption fulfilled only approximately, etc.

As usual for any mathematical logic, in fuzzy logic we clearly distinguish syntax from semantics. While semantics is always many-valued, based on a certain algebraic structure of truth values, syntax can be either traditional or evaluated. In comparison with classical logic, fuzzy logic has different and wider repertoire of connectives and logical axioms but similar inference rules. All known kinds of fuzzy logic with traditional syntax including first-order are complete that is, a formula is provable if and only if it is true in the degree one in all models.

A more radical departure from classical logic is fuzzy logic with evaluated syntax (J. Pavelka, V. Novák). Its fundamental concept is evaluated formula a/A where A is a formula and a is its syntactic evaluation that is, a lower bound for the truth of A in any model. This makes it possible to consider axioms that need not be fully convincing. The concepts such as evaluated proof, provability degree and some other ones are introduced. Important result is a generalization of the Gödel completeness theorem: the provability degree of an arbitrary formula in arbitrary fuzzy theory is equal to its truth degree. It has been proved that such generalization is possible only in case that the structure of truth values is isomorphic with the standard or finite Łukasiewicz algebra.

2. Focus of our research

IRAFM has focused on the following topics:

• Formal systems of fuzzy logic in narrow sense:
• fuzzy logic with evaluated syntax,
• higher order fuzzy logic (fuzzy type theory),
• Model theory for fuzzy logic including models of higher order fuzzy logic,
• Model theory of fuzzy logic in categories,
• Theory of generalized quantifiers, nonstandard extensions of fuzzy logic,
• Category of fuzzy sets and sets with generalized equality as a basis for models of fuzzy logic.

3. Description of the main results

1. We have developed in detail calculus of predicate first-order fuzzy logic with evaluated syntax. The results are summarized in the book [47]. Further results including model theory of this logic are presented in [3], [48], [28], [32], [34], [36], [37], [38], [49], [39], [45]. In model theory we have generalized Robinson-Craig theorem on union of fuzzy theories [29] and omitting types theorem [23], [24], [26]. Results concerning automation of provability in this logic are contained in [5], [6], [7], [8].
2. We have developed full calculus of fuzzy type theory that is, a higher-order fuzzy logic of Henkin type. The fundamental paper is [33] where the considered structure of truth values is IMTL-algebra. Further extension is contained in [35] (extension by description operator). Fuzzy type theory for Łukasiewicz, BL and ŁΠ algebras of truth values is presented in [1], [41], [42]. The completeness theorem for all the above mentioned kinds of fuzzy type theory with respect to general models has been proved.
3. We have described special categories, namely categories of fuzzy sets with values in MV-algebra, studied their morphisms and subobjects with respect to interpretation of fuzzy logic in such categories [14], [15], [16], [17], [19], [20]. The other special categories are categories of fuzzy sets with generalized equality.
4. We have extended fuzzy logic by the theory of generalized quantifiers as a direct generalization of classical theory of generalized quantifiers [9], [10] , [11].

REFERENCES:

[1] DVOŘÁK, A., NOVÁK, V. Fuzzy Type Theory as a Tool for Linguistic Analysis. In The Logica Yearbook 2006. Prague : Filosofia, 2007. ISBN 80-7007-254-7. pp. 51-61.

[2] DVOŘÁK, A., NOVÁK, V. Fuzzy Logic as a Methodology for the Treatment of Vagueness. In The logica yearbook 2004. Praha : Filosofia, 2005. ISBN 80-7007-208-3. pp. 141-152.

[3] GOTTWALD, S., V. NOVÁK: An Approach Towards Consistency Degrees of Fuzzy Theories. Int. J. of General Systems, 29(2000), 499–510.

[4] HÁJEK, P., NOVÁK, V. The Sorites paradox and fuzzy logic. In J. of General Systems. 2003, vol.32, pp.373-383, ISSN 0308-1079.

[5] HABIBALLA, H. Non-clausal Resolution Principle for Fuzzy Description Logic. In INFORMATICA, ISSN 0868-4952.

[6] HABIBALLA, H.: Resolution Strategies for Fuzzy Description Logic. In EUSFLAT. 10.9.2007-14.9.2007 Ostrava. Ostrava: 2007.

[7] HABIBALLA, H. Non-clausal Resolution Theorem Proving for Fuzzy Description Logic. In SOFSEM 2006. 2006-01-21-2006-01-27 Měřín. Praha :Institute of Computer Science, Czech Academy of Sciences, 2006. pp. 1-12. ISBN 80-903298-4-5 .

[8] HABIBALLA, H. Non-clausal Resolution in Fuzzy Predicate Logic with Evaluated Syntax (background and implementation). In The Logic of Soft Computing IV. 5.10.2005-7.10.2005 Ostrava. Ostrava : Ostravská Univerzita, 2005. pp. 51-54.

[9] HOLČAPEK, M. An axiomatic approach to cardinalities of finite L-fuzzy sets. In Fuzzy sets and Systems. 2008 , ISSN 0165-0114.

[10] HOLČAPEK, M. Monadic L-fuzzy quantifiers of the type <1n,1>. In Fuzzy sets and Systems. 2008, ISSN 0165-0114.

[11] HOLČAPEK, M. Fuzzy logic with fuzzy quantifiers. In IPMU 2006 (Information Processing and Management of Uncertainty in Knowledge-based Systems). 2006-07-02-2006-07-07, Paris, France. Paris : Paris: Editions E.D.K., 2006. pp. 1882-1889. ISBN 2-84254-112-X.

[12] ILIRINNE, E., TURUNEN, E.: GUHA Data Mining Quantifiers Interpreted in Mathematical Fuzzy Logic. Fuzzy Sets and Systems (submitted).

[13] MOČKOŘ, J. Fuzzy and non-deterministic automata. In Soft Computing. 1999, 3, pp.221-226, ISSN 1432-7643.

[14] MOČKOŘ, J. Covariant functors in categories of fuzzy sets over MV-algebras. Fuzzy Sets and Systems. 1(2). 2006, 2, pp.83-109, ISSN 0973-421X .

[15] MOČKOŘ, J. Extensional subobjects in categories of Omega-fuzzy sets. In Czechoslovak Mathematical Journal. 2006, pp.1-10, ISSN 1211-4774.

[16] MOČKOŘ, J.: Complete subobject of fuzzy sets over MV-algebras, Czechoslovak Mathematical Journal. 2004, vol.129, 54, pp.379-392, ISSN 1211-4774

[17] MOČKOŘ, J. Fuzzy logic models in a category of sets with similarity relations. In Czech-Japan Seminar on Fuzzy Systems and Inovational Computing. 2006-08-18-2006-08-22, Fukuoka, Japan. Fukuoka : Waseda University, 2006. pp. 58-64.

[18] MOČKOŘ, J. Extension principle for category of fuzzy sets over MV-algebras. In Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty. Mt. Koyasan, Japan. Koyasan: University of Osaka, 2002. pp. 160-166. ISBN 1111-1111.

[19] MOČKOŘ, J. Extensional objects and complete sets in categories of fuzzy sets over MV-algebras. In 8th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty. 2005-09-18-2005-09-21 Třešť. Praha : VŠE Praha, 2005. pp. 76-81. ISBN 80-245-0915-6.

[20] MOČKOŘ, J. Weak reflections in categories of fuzzy sets over MV-algebras. In The logic of soft computing IV, 4th Workshop of the ERCIM Working group on soft computing. 5.10.2005-7.10.2005, Ostrava. Ostrava : Universzity of Ostrava, 2005. pp. 88-89.

[21] MOČKOŘ, J., SMOLÍKOVÁ, R.: Fuzzy automata. In Tatra Mountains Mathematical Publications. 1997, vol.12, 2, pp.41-50, ISSN 1210-3195.

[22] MOČKOŘ, J., SMOLÍKOVÁ, R.: Category of extended fuzzy automata. In Acta Mathematica at Informatica Universitatis Ostraviensis. 1996, vol.1, 4, pp.47-56, ISSN 1211-4774.

[23] MURINOVÁ, P.: The Omitting Types in predicate fuzzy logics, Journal of Electrical Engineering, Vol. 55, No. 12/s, Bratislava, 2004, 87-90.

[24] MURINOVÁ, P. Fuzzy logic with evaluated syntax extended by product. In Journal of Electrical Engeneering. 12. 2003, 54, pp.85-88, ISSN 1335-3632.

[25] MURINOVÁ, P. Model theory in fuzzy logic with evaluated syntax extended by product. In Journal of electrical engeneering. 12. 2003, 54, pp.86-89, ISSN 1335-3632.

[26] MURINOVÁ, P., NOVÁK, V. Omitting Types in Fuzzy Logic with Evaluated Syntax. In Mathematical Logic Quarterly. 2006, vol.52, pp.259-268, ISSN 0942-5616.

[27] NOVÁK, V. Which logic is the real fuzzy logic?. In Fuzzy Sets and Systems. 2006,157, 5, pp.635-641, ISSN 0165-0114.

[28] NOVÁK, V. Fuzzy logic with countable evaluated syntax revisited. In Fuzzy Sets and Systems., 2007, vol.158, 1, pp.929-936, ISSN 0165-0114.

[29] NOVÁK, V. Joint consistency of fuzzy Theories. In Mathematical Logic Quaterly. 2002, vol.48, pp.563-573, ISSN 0942-5616.

[30] NOVÁK, V. Intensional Theory of Granular Computing. In Soft Computing. 2004, vol.8, pp.281-290, ISSN 1432-7643.

[31] NOVÁK, V. Antonyms and Linguistic Quantifiers in Fuzzy Logic. In Fuzzy Sets and Systems. 2001, vol.124, pp.335-351, ISSN 0165-0114.

[32] NOVÁK, V. Are Fuzzy Sets a Reasonable Tool for Modeling Vague Phenomena?. In Fuzzy Sets and Systems. 2005,156, pp.341-348, ISSN 0165-0114.

[33] NOVÁK, V. Paradigm, Formal Properties and Limits of Fuzzy Logic. In Int. J. of General Systems. 1996, 24, pp.377-405, ISSN 0308-1079.

[34] NOVÁK, V. On Fuzzy Type Theory. In Fuzzy Sets and Systems. 2005, vol.149, pp.235-273, ISSN 0165-0114.

[35] NOVÁK, V. Open Theories, Consistency and Related Results in Fuzzy Logic. In Int. Journal of Approximate Reasoning. 1998, 18, pp.191-200, ISSN 0888-613X.

[36] NOVÁK, V. Descriptions in Full Fuzzy Type Theory. In Neural Network World. 2003,5, 13, pp.559-569, ISSN 1210-0552.

[37] NOVÁK, V. Fuzzy Functions in Fuzzy Logic with Fuzzy Equality. In Acta Mathematica et Informatica Universitatis Ostraviensis. 2001, 9, pp.59-66, ISSN 1211-4774.

[38] NOVÁK, V. On functions in fuzzy logic with evaluated syntax. In Neural network World. 2. 2000, vol.10, 10, pp.869-875, ISSN 1210-0552.

[39] NOVÁK, V. On fuzzy equality and approximation in fuzzy logic. In Soft Computing. 2004, vol.8, 10, pp.668-675, ISSN 1432-7643.

[40] NOVÁK, V. On the Hilbert—Ackermann Theorem in Fuzzy Logic. In Acta Mathematica et Informatica Universitatis Ostraviensis. 1996, 4, pp.57-74, ISSN 1211-4774.

[41] NOVÁK, V. Models and submodels of fuzzy Theories. In IPMU 2002. 20020701-20020705 Annecy. Annecy : ESIA-Université de Savoie, 2002. pp. 385-390. ISBN 2-9516453-1-7.

[42] NOVÁK, V. From Fuzzy Type Theory to Fuzzy Intensional Logic. In Third Conf. EUSFLAT 2003. 10. 9. 2003-12. 9. 2003 Zittau/Goerlitz, Německo. Zittau/Goerlitz : University of Applied Science at Zittau/Goerlitz , 2003. pp. 619-623. ISBN 3-9808089-4-7.

[43] NOVÁK, V. Fuzzy Type Theory As Higher Order Fuzzy Logic. In The Sixth International Conference on Intelligent Technologies. 14. 12. 2005-16. 12. 2005 Phuket, Thailand. Bankgok: Faculty of Science and Technology, Assumption University, Bangkok, Thailand, 2005. pp. 21-26. ISBN 974-615-226-2.

[44] NOVÁK, V. Fuzzy Logic Theory of Evaluating Expressions and Comparative Quantifiers. In IMPU'06. 2006-07-02-2006-07-05 Paris. Paris : Editions EDK, 2006. pp. 1572-1579. ISBN 2-84254-112-X.

[45] NOVÁK, V. A Formal Theory of Intermediate Quantifiers. In Fuzzy Sets and Systems, ISSN 0165-0114. (to appear)

[46] NOVÁK, V., MURINOVÁ, P.: Omitting Types in Fuzzy Predicate Logics. S. Gottwald, P. Hájek, U. Höhle, E. P. Klement (Eds.), Proc. of 26th Linz Seminar on Fuzzy Set Theory, (to appear)

[47] NOVÁK, V., PERFILIEVA, I. Discovering the World with Fuzzy Logic. Heidelberg: Springer-Verlag, 2000. 555 pp. Studies in Fuzziness and Soft Computing. ISBN 3-7908-1330-3.

[48] NOVÁK, V., PERFILIEVA, I., MOČKOŘ, J. Mathematical Principles of Fuzzy Logic. Boston/Dordrecht/London : Kluwer Academic publishers, 1999. 320 pp. ISBN 0-7923-8595-0.

[49] NOVÁK, V., PERFILIEVA, I. The principles of fuzzy logic: its mathematical and computational aspects. In Lectures on Soft Computing. Heidelberg : Springer, 2001. ISBN 3-7908-1396-6. pp. 189-238.

[50] PERFILIEVA, I., NOVÁK, V. Some Consequences of Herbrand and McNaughton Theorems in Fuzzy Logic. In Discovering World With Fuzzy Logic. Heidelberg: Springer-Verlag, 2000. ISBN 3-7908-1330-3. pp. 271-295.

Updated: 16. 05. 2018